given cost and revenue functions $c(x)=5x + 75$ and $r(x)=240x - 0.1x^{2}$, find the rate of change of the…

Explanation:

Step1: Define the profit function

The profit function $P(x)$ is given by $P(x)=R(x)-C(x)$. So, $P(x)=(240x - 0.1x^{2})-(5x + 75)=- 0.1x^{2}+240x - 5x-75=-0.1x^{2}+235x - 75$.

Step2: Differentiate the profit function

Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we differentiate $P(x)$ with respect to $x$. For $y=-0.1x^{2}$, $y^\prime=-0.1\times2x=-0.2x$. For $y = 235x$, $y^\prime=235$. For $y=-75$, $y^\prime = 0$. So, $P^\prime(x)=-0.2x + 235$.

Answer:

$-0.2x + 235$